degrees, and BAC to 41, and the triangle will be formed; then take in your compasses the length of the side A C, and apply it to the same scale, and you will find its length to be 24 chains; do the same by the side B C, and you will find it measure 27 chains, and the protractor will shew that the angle A CB contains 103 degrees.

Example 6. Given the base A B, fig. 16, plate 3, of a triangle 327 yards, the angle CAB 44° 30', and the side AC, 208 yards, to constitute the said triangle, and find the length of the other sides.

Draw the line AB at pleasure, then take 327 parts from the scale, and lay it from A to B ; set the centre of the protractor to the point A, lay off 44° 30', and by that mark draw AC; then take with the compasses from the same scale 208, and lay it from A to C, and join CB, and the parts of the triangle in the plan will bear the same proportion to each other as the real parts in the field do.

OF THE REMAINING LINES ON THE PLAIN
SCALE.

OF THE PROTRACTING SCALES.

1. Of the line of chords. This line is used to set off an angle from a given point in any right line, or to measure the quantity of an angle already laid down.

Thus to draw a line Ae, fig. 14, plate 3, that shall make with the line AB an angle, containing a given number of degrees, suppose 40 degrees.

Open your compasses to the extent of 60 degrees upon the line of chords. (which is always equal to

the radius of the circle of projection,) and setting one foot in the angular point, with that extent describe an arch; then taking the extent of 40 degrees from the said chord line, set it off from the given line on the arch described at e; a right line drawn from the given point, through the point marked upon the arch, will form the required angle. The degrees contained in an angle already laid down, are found nearly in the same manner; for instance, to measure the angle CAB. From the centre describe an arch with the chord of 60 degrees, and the distance CB, measured on the line of chords, will give the number of degrees contained in the angle.

If the number of degrees are more than' 90, they must be measured upon the chords at twice: thus, if 120 degrees were to be practised, 60 may be taken from the chords, and those degrees be laid off twice upon the arch. Degrees taken from the chords are always to be counted from the beginning

of the scale.

Of the rhumb line. This is, in fact, a line of chords constructed to a quadrant divided into eight parts or points of the compass, in order to facilitate the work of the navigator in laying down a ship's

course.

Thus, supposing the ship's course to be NNE, and it be required to lay down that angle. Draw the line AB, fig. 14, plate 3, to represent the meridian, and about the point A sweep an arch with the chord of 60 degrees; then take the extent to the second rhumb, from the line of rhumbs, and set it off on the arch from B to e, and draw the line A e, and the angle B Ae will represent the ship's course. The second rhumb was taken, because NNE is the second point from the north.

Of the line of longitudes. The line of longitudes is a line divided into sixty unequal parts, and so applied to the line of chords, as to shew by inspection,

the number of equatorial miles contained in a degree on any parallel of latitude. The graduated line of chords is necessary, in order to shew the latitudes; the line of longitude shews the quantity of a degree on each parallel in sixtieth parts of an equa torial degree, that is, miles. The use of this line will be evident from the following example. A ship in latitude 44° 12′ N. sails E. 79 miles, required her difference of longitude. Opposite to 44+ nearest equal to the latitude on the line of chords, stands 43 on the line of longitude, which is therefore the number of miles in a degree of longitude in that latitude. Whence as 43: 60 :: 79: 110 miles, the difference of longitude.

The lines of tangents, semitangents, and secants serve to find the centres and poles of projected circles in the stereographical projection of the sphere,

The line of sines is principally used for the orthographic projection of the sphere; but as the application of these lines is the same as that of similar lines on the sector, we shall refer the reader to the explanation of those lines in our description of that instrument.

The lines of latitudes and hours are used conjointly, and serve very readily to mark the hour lines in the construction of dials; they are generally on the most complete sorts of scales and sectors; for the uses of which see treatises on dialling.

* I think it however best to give the young student here the simple method of drawing the lines for a dial by the common scales of latitudes and hours. It is the most ready way, and when the lines are correctly laid down, on an extent from 9 to 12 inches or more, the hour lines of the dial may be delineated even to a minute of time. For an horizontal dial, plate4, figs. 19, 19*. Draw two parallel lines, as a double nieridian line a b c d, at a distance apart equal to the thickness of the intended stile or gnomon, on your dial plane. Intersect it at right angles by another line, called the 6 o'clock line, ef, from the scale of latitudes take the latitude of the place with the compasses, and set that extent from c to e and from a to f on the 6 o'clock line, then taking the whole 6 hours between the parts of the compasses from the scale, with

OF THE SECTOR.

Amidst the variety of mathematical instruments that have been contrived to facilitate the art of drawing, there is none so extensive in its use, or of such general application as the sector. It is an

that extent set one foot in the point e, and with the other intersect the meridian line cd at d. Do the same from f to h, and draw the right lines c dfb, which are of the same length as the scale of hours. Place one foot of the compasses in the beginning of the scale at XII, and extending the other to any hour on the scale, lay these extents off from d towards e for the afternoon hours, and from towards f for the forenoon. In the same manner the quarters or minutes may be laid down if required The edge of a ruler being placed on the point c, draw the first five afternoon hours from that point through the marks 1, 2, 3, 4, 5, on the line de, and continue the lines of IV and V through the centre c to the other side of the dial for the like hours of the morning: lay a ruler on the point a and draw the last five forenoon hours, through the tharks 5, 4, 3, 2, 1, as the line fb continuing the hour lines of VII and VIII through the centre a to the other side of the dial, for the evening hours, and figure the hours to the respective lines. To make the gnomon. From the line of chords always placed dialling scale take the extent of 60°, and describe from the centre a the arch ge, then with the extent of the latitude of the place, suppose London 51° taken from the line of chords, set one foot in a and cross the arch with the other at n. From the centre at a draw the line a g for the axis of the gnomon 4g, and from g let fall the perpendicular g i upon the meridian line ai, and there will be formed a triangle a gi for a plate or triangular frame similar to this triangle, of the thickness of the interval of the parallel lines ac bd which must be made and set upright between them touching at a and 6: its hypothenuse or axis a g will be parallel to the axis of the earth when the dial is fixed truly, and of course cast its shadow on the hour of the day. To make an erect south dial, take the complement of the latitude of the place, which for London, is 90° less 51+= 38 from the scale of latitudes, and proceed in all respects for the hour lines, as above for the horizontal dial, only reversing the hours, and limiting them to the VI, and for the gnomon making the angle of the stile's height equal to the co-latitude 384, see fig. 20.

on the same

Lines of chords, and inclinations of meridians, are used when the hour angles are given in tables of degrees, &c.

For the easiest and best methods of fixing dials, see my Method of finding a Meridian Line to set Sun Dials, Clocks, Watches, &c. SFO. EDIT.

universal scale, uniting as it were, angles and parallel lines, the rule and the compass, which are the only means that geometry makes use of for measuring, whether in speculation or practice. The real inventor of this valuable instrument is unknown; yet of so much merit has the invention appeared, that it was claimed by Galileo, and disputed by

nations.

This instrument derives its name from the tenth definition of the third book of Euclid, where he defines the sector of a circle. It is formed of two equal rules, (fig. 4 and 5, plate 3,) A B, DB, called legs; these legs are moveable about the centre C of a joint de f, and will, consequently, by their different openings, represent every possible variety of plane angles. The distance of the extremities of these rules are the subtenses or chords, of the arches they describe.

Sectors are made of different sizes, but their length is usually denominated from the length-of the legs when the sector is shut. Thus a sector of six inches, when the legs are close together, forms a rule of 12 inches when opened; and a foot sector is two feet long, when opened to its greatest extent. In describing the lines usually placed on this instrument, I refer to those commonly laid down on the best six-inch brass sectors. But as the principles are the same in all, and the differences little more than in the number of subdivisions, it is to be presumed that no difficulty will occur in the application of what is here said to sectors of a larger radius.

Of this instrument, Dr. Priestley thus speaks in his Treatise on Perspective. "Besides the small sector in the common pocket cases of instruments, I would advise a person who proposes to learn to draw, to get another of one foot radius. Two sectors are in many cases exceeding useful, if not absolutely necessary; and I would not advise a